Calculus Examples

$\int - 2 (5 x^{2} + 2 x) e^{3 x} d x$

Step 1

Simplify.

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Step 1.1

Apply the distributive property.

$\int (- 2 (5 x^{2}) - 2 (2 x)) e^{3 x} d x$

Step 1.2

Multiply $5$ by $- 2$ .

$\int (- 10 x^{2} - 2 (2 x)) e^{3 x} d x$

Step 1.3

Multiply $2$ by $- 2$ .

$\int (- 10 x^{2} - 4 x) e^{3 x} d x$

Step 1.4

Apply the distributive property.

$\int - 10 x^{2} e^{3 x} - 4 x e^{3 x} d x$

Step 2

Split the single integral into multiple integrals.

$\int - 10 x^{2} e^{3 x} d x + \int - 4 x e^{3 x} d x$

Step 3

Since $- 10$ is constant with respect to $x$ , move $- 10$ out of the integral.

$- 10 \int x^{2} e^{3 x} d x + \int - 4 x e^{3 x} d x$

Step 4

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x^{2}$ and $d v = e^{3 x}$ .

$- 10 (x^{2} (\frac{1}{3} e^{3 x}) - \int \frac{1}{3} e^{3 x} (2 x) d x) + \int - 4 x e^{3 x} d x$

Step 5

Simplify.

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Step 5.1

Combine $\frac{1}{3}$ and $e^{3 x}$ .

$- 10 (x^{2} \frac{e^{3 x}}{3} - \int \frac{1}{3} e^{3 x} (2 x) d x) + \int - 4 x e^{3 x} d x$

Step 5.2

Combine $x^{2}$ and $\frac{e^{3 x}}{3}$ .

$- 10 (\frac{x^{2} e^{3 x}}{3} - \int \frac{1}{3} e^{3 x} (2 x) d x) + \int - 4 x e^{3 x} d x$

Step 5.3

Combine $\frac{1}{3}$ and $e^{3 x}$ .

$- 10 (\frac{x^{2} e^{3 x}}{3} - \int \frac{e^{3 x}}{3} (2 x) d x) + \int - 4 x e^{3 x} d x$

Step 5.4

Combine $2$ and $\frac{e^{3 x}}{3}$ .

$- 10 (\frac{x^{2} e^{3 x}}{3} - \int \frac{2 e^{3 x}}{3} x d x) + \int - 4 x e^{3 x} d x$

Step 5.5

Combine $\frac{2 e^{3 x}}{3}$ and $x$ .

$- 10 (\frac{x^{2} e^{3 x}}{3} - \int \frac{2 e^{3 x} x}{3} d x) + \int - 4 x e^{3 x} d x$

Step 6

Since $\frac{2}{3}$ is constant with respect to $x$ , move $\frac{2}{3}$ out of the integral.

$- 10 (\frac{x^{2} e^{3 x}}{3} - (\frac{2}{3} \int e^{3 x} x d x)) + \int - 4 x e^{3 x} d x$

Step 7

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x$ and $d v = e^{3 x}$ .

$- 10 (\frac{x^{2} e^{3 x}}{3} - \frac{2}{3} (x (\frac{1}{3} e^{3 x}) - \int \frac{1}{3} e^{3 x} d x)) + \int - 4 x e^{3 x} d x$

Step 8

Simplify.

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Step 8.1

Combine $\frac{1}{3}$ and $e^{3 x}$ .

$- 10 (\frac{x^{2} e^{3 x}}{3} - \frac{2}{3} (x \frac{e^{3 x}}{3} - \int \frac{1}{3} e^{3 x} d x)) + \int - 4 x e^{3 x} d x$

Step 8.2

Combine $x$ and $\frac{e^{3 x}}{3}$ .

$- 10 (\frac{x^{2} e^{3 x}}{3} - \frac{2}{3} (\frac{x e^{3 x}}{3} - \int \frac{1}{3} e^{3 x} d x)) + \int - 4 x e^{3 x} d x$

Step 8.3

Combine $\frac{1}{3}$ and $e^{3 x}$ .

$- 10 (\frac{x^{2} e^{3 x}}{3} - \frac{2}{3} (\frac{x e^{3 x}}{3} - \int \frac{e^{3 x}}{3} d x)) + \int - 4 x e^{3 x} d x$

Step 9

Since $\frac{1}{3}$ is constant with respect to $x$ , move $\frac{1}{3}$ out of the integral.

$- 10 (\frac{x^{2} e^{3 x}}{3} - \frac{2}{3} (\frac{x e^{3 x}}{3} - (\frac{1}{3} \int e^{3 x} d x))) + \int - 4 x e^{3 x} d x$

Step 10

Let $u_{1} = 3 x$ . Then $d u_{1} = 3 d x$ , so $\frac{1}{3} d u_{1} = d x$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

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Step 10.1

Let $u_{1} = 3 x$ . Find $\frac{d u_{1}}{d x}$ .

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Step 10.1.1

Differentiate $3 x$ .

$\frac{d}{d x} [3 x]$

Step 10.1.2

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$3 \frac{d}{d x} [x]$

Step 10.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$3 \cdot 1$

Step 10.1.4

Multiply $3$ by $1$ .

$3$

Step 10.2

Rewrite the problem using $u_{1}$ and $d u_{1}$ .

$- 10 (\frac{x^{2} e^{3 x}}{3} - \frac{2}{3} (\frac{x e^{3 x}}{3} - \frac{1}{3} \int e^{u_{1}} \frac{1}{3} d u_{1})) + \int - 4 x e^{3 x} d x$

Step 11

Combine $e^{u_{1}}$ and $\frac{1}{3}$ .

$- 10 (\frac{x^{2} e^{3 x}}{3} - \frac{2}{3} (\frac{x e^{3 x}}{3} - \frac{1}{3} \int \frac{e^{u_{1}}}{3} d u_{1})) + \int - 4 x e^{3 x} d x$

Step 12

Since $\frac{1}{3}$ is constant with respect to $u_{1}$ , move $\frac{1}{3}$ out of the integral.

$- 10 (\frac{x^{2} e^{3 x}}{3} - \frac{2}{3} (\frac{x e^{3 x}}{3} - \frac{1}{3} (\frac{1}{3} \int e^{u_{1}} d u_{1}))) + \int - 4 x e^{3 x} d x$

Step 13

Simplify.

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Step 13.1

Multiply $\frac{1}{3}$ by $\frac{1}{3}$ .

$- 10 (\frac{x^{2} e^{3 x}}{3} - \frac{2}{3} (\frac{x e^{3 x}}{3} - \frac{1}{3 \cdot 3} \int e^{u_{1}} d u_{1})) + \int - 4 x e^{3 x} d x$

Step 13.2

Multiply $3$ by $3$ .

$- 10 (\frac{x^{2} e^{3 x}}{3} - \frac{2}{3} (\frac{x e^{3 x}}{3} - \frac{1}{9} \int e^{u_{1}} d u_{1})) + \int - 4 x e^{3 x} d x$

Step 14

The integral of $e^{u_{1}}$ with respect to $u_{1}$ is $e^{u_{1}}$ .

$- 10 (\frac{x^{2} e^{3 x}}{3} - \frac{2}{3} (\frac{x e^{3 x}}{3} - \frac{1}{9} (e^{u_{1}} + C))) + \int - 4 x e^{3 x} d x$

Step 15

Since $- 4$ is constant with respect to $x$ , move $- 4$ out of the integral.

$- 10 (\frac{x^{2} e^{3 x}}{3} - \frac{2}{3} (\frac{x e^{3 x}}{3} - \frac{1}{9} (e^{u_{1}} + C))) - 4 \int x e^{3 x} d x$

Step 16

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x$ and $d v = e^{3 x}$ .

$- 10 (\frac{x^{2} e^{3 x}}{3} - \frac{2}{3} (\frac{x e^{3 x}}{3} - \frac{1}{9} (e^{u_{1}} + C))) - 4 (x (\frac{1}{3} e^{3 x}) - \int \frac{1}{3} e^{3 x} d x)$

Step 17

Simplify.

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Step 17.1

Combine $\frac{1}{3}$ and $e^{3 x}$ .

$- 10 (\frac{x^{2} e^{3 x}}{3} - \frac{2}{3} (\frac{x e^{3 x}}{3} - \frac{1}{9} (e^{u_{1}} + C))) - 4 (x \frac{e^{3 x}}{3} - \int \frac{1}{3} e^{3 x} d x)$

Step 17.2

Combine $x$ and $\frac{e^{3 x}}{3}$ .

$- 10 (\frac{x^{2} e^{3 x}}{3} - \frac{2}{3} (\frac{x e^{3 x}}{3} - \frac{1}{9} (e^{u_{1}} + C))) - 4 (\frac{x e^{3 x}}{3} - \int \frac{1}{3} e^{3 x} d x)$

Step 17.3

Combine $\frac{1}{3}$ and $e^{3 x}$ .

$- 10 (\frac{x^{2} e^{3 x}}{3} - \frac{2}{3} (\frac{x e^{3 x}}{3} - \frac{1}{9} (e^{u_{1}} + C))) - 4 (\frac{x e^{3 x}}{3} - \int \frac{e^{3 x}}{3} d x)$

Step 18

Since $\frac{1}{3}$ is constant with respect to $x$ , move $\frac{1}{3}$ out of the integral.

$- 10 (\frac{x^{2} e^{3 x}}{3} - \frac{2}{3} (\frac{x e^{3 x}}{3} - \frac{1}{9} (e^{u_{1}} + C))) - 4 (\frac{x e^{3 x}}{3} - (\frac{1}{3} \int e^{3 x} d x))$

Step 19

Let $u_{2} = 3 x$ . Then $d u_{2} = 3 d x$ , so $\frac{1}{3} d u_{2} = d x$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 19.1

Let $u_{2} = 3 x$ . Find $\frac{d u_{2}}{d x}$ .

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Step 19.1.1

Differentiate $3 x$ .

$\frac{d}{d x} [3 x]$

Step 19.1.2

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$3 \frac{d}{d x} [x]$

Step 19.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$3 \cdot 1$

Step 19.1.4

Multiply $3$ by $1$ .

$3$

Step 19.2

Rewrite the problem using $u_{2}$ and $d u_{2}$ .

$- 10 (\frac{x^{2} e^{3 x}}{3} - \frac{2}{3} (\frac{x e^{3 x}}{3} - \frac{1}{9} (e^{u_{1}} + C))) - 4 (\frac{x e^{3 x}}{3} - \frac{1}{3} \int e^{u_{2}} \frac{1}{3} d u_{2})$

Step 20

Combine $e^{u_{2}}$ and $\frac{1}{3}$ .

$- 10 (\frac{x^{2} e^{3 x}}{3} - \frac{2}{3} (\frac{x e^{3 x}}{3} - \frac{1}{9} (e^{u_{1}} + C))) - 4 (\frac{x e^{3 x}}{3} - \frac{1}{3} \int \frac{e^{u_{2}}}{3} d u_{2})$

Step 21

Since $\frac{1}{3}$ is constant with respect to $u_{2}$ , move $\frac{1}{3}$ out of the integral.

$- 10 (\frac{x^{2} e^{3 x}}{3} - \frac{2}{3} (\frac{x e^{3 x}}{3} - \frac{1}{9} (e^{u_{1}} + C))) - 4 (\frac{x e^{3 x}}{3} - \frac{1}{3} (\frac{1}{3} \int e^{u_{2}} d u_{2}))$

Step 22

Simplify.

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Step 22.1

Multiply $\frac{1}{3}$ by $\frac{1}{3}$ .

$- 10 (\frac{x^{2} e^{3 x}}{3} - \frac{2}{3} (\frac{x e^{3 x}}{3} - \frac{1}{9} (e^{u_{1}} + C))) - 4 (\frac{x e^{3 x}}{3} - \frac{1}{3 \cdot 3} \int e^{u_{2}} d u_{2})$

Step 22.2

Multiply $3$ by $3$ .

$- 10 (\frac{x^{2} e^{3 x}}{3} - \frac{2}{3} (\frac{x e^{3 x}}{3} - \frac{1}{9} (e^{u_{1}} + C))) - 4 (\frac{x e^{3 x}}{3} - \frac{1}{9} \int e^{u_{2}} d u_{2})$

Step 23

The integral of $e^{u_{2}}$ with respect to $u_{2}$ is $e^{u_{2}}$ .

$- 10 (\frac{x^{2} e^{3 x}}{3} - \frac{2}{3} (\frac{x e^{3 x}}{3} - \frac{1}{9} (e^{u_{1}} + C))) - 4 (\frac{x e^{3 x}}{3} - \frac{1}{9} (e^{u_{2}} + C))$

Step 24

Simplify.

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Step 24.1

Simplify.

$- 10 (\frac{x^{2} e^{3 x}}{3} - \frac{2 x e^{3 x}}{9} + \frac{2 e^{u_{1}}}{27}) - 4 (\frac{x e^{3 x}}{3} - \frac{1}{9} e^{u_{2}}) + C$

Step 24.2

Combine $e^{u_{2}}$ and $\frac{1}{9}$ .

$- 10 (\frac{x^{2} e^{3 x}}{3} - \frac{2 x e^{3 x}}{9} + \frac{2 e^{u_{1}}}{27}) - 4 (\frac{x e^{3 x}}{3} - \frac{e^{u_{2}}}{9}) + C$

Step 25

Substitute back in for each integration substitution variable.

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Step 25.1

Replace all occurrences of $u_{1}$ with $3 x$ .

$- 10 (\frac{x^{2} e^{3 x}}{3} - \frac{2 x e^{3 x}}{9} + \frac{2 e^{3 x}}{27}) - 4 (\frac{x e^{3 x}}{3} - \frac{e^{u_{2}}}{9}) + C$

Step 25.2

Replace all occurrences of $u_{2}$ with $3 x$ .

$- 10 (\frac{x^{2} e^{3 x}}{3} - \frac{2 x e^{3 x}}{9} + \frac{2 e^{3 x}}{27}) - 4 (\frac{x e^{3 x}}{3} - \frac{e^{3 x}}{9}) + C$

Step 26

Reorder terms.

$- 10 (\frac{1}{3} x^{2} e^{3 x} - \frac{2}{9} x e^{3 x} + \frac{2}{27} e^{3 x}) - 4 (\frac{1}{3} x e^{3 x} - \frac{1}{9} e^{3 x}) + C$